Definition

Let ${\displaystyle X}$ be a set. Then a metric on the set ${\displaystyle X}$ is a function ${\displaystyle d:X\times X\to \mathbb {R} }$ such that

1. ${\displaystyle d(x,y)\geq 0}$ and equality holds if and only if ${\displaystyle x=y.}$
2. ${\displaystyle d(x,y)=d(y,x)}$ (i.e., ${\displaystyle d}$ is symmetric).
3. ${\displaystyle d(x,z)\leq d(x,y)+d(y,z)}$ (the triangle inequality).

We can define a topology on the set ${\displaystyle X}$ using the metric ${\displaystyle d}$ as follows. For ${\displaystyle x\in X}$ and ${\displaystyle r>0}$ define ${\displaystyle B(x,r)=\{y\in X\mid d(x,y)<r\}.}$ Then take the collection ${\displaystyle \{B(x,r)\mid x\in X,\ r>0\}}$ as a basis. Call this topology ${\displaystyle {\mathcal {T}}_{d}.}$

If ${\displaystyle (X,{\mathcal {T}})}$ is a topological space, then we say that it is metrizable if there is a metric ${\displaystyle d:X\times X\to \mathbb {R} }$ such that ${\displaystyle {\mathcal {T}}={\mathcal {T}}_{d}.}$ If ${\displaystyle X}$ is metrizable and ${\displaystyle d}$ is such a metric then we call the pair ${\displaystyle (X,d)}$ a metric space.

Examples

1. The most obvious example is the space ${\displaystyle \mathbb {R} }$ together with the metric ${\displaystyle d(x,y)=|x-y|}$ which is called the standard metric on ${\displaystyle \mathbb {R} .}$
2. Any set has a metric on it. As an example, let ${\displaystyle X}$ be any set and let ${\displaystyle d}$ be given by ${\displaystyle d(x,y)=\left\{{\begin{array}{ll}1&x\neq y\\0&x=y\end{array}}\right..}$ It is left as an exercise to check that this is a metric, called the discrete metric because the topology it gives ${\displaystyle X}$ is the discrete topology.

Exercises

1. Show that the discrete metric is a metric and that it gives the discrete topology on ${\displaystyle X.}$

Definition

Let ${\displaystyle (X,d)}$ and ${\displaystyle (Y,\rho )}$ be metric spaces. Then a function ${\displaystyle f:X\to Y}$ is called an isometry if for all ${\displaystyle x,y\in X}$ we have ${\displaystyle \rho (f(x),f(y))=d(x,y)}$ (i.e. if distances are preserved). If ${\displaystyle f}$ is also surjective, then we say that ${\displaystyle f}$ is a global isometry.

Examples

1. If ${\displaystyle (X,d)}$ is any metric space, then the identity map is an isometry.
2. If ${\displaystyle (Y,d)}$ is a metric space and ${\displaystyle X\subset Y}$ then the inclusion map ${\displaystyle i:X\to Y}$ is an isometry.

Exercises

1. Show that every isometry is injective and therefore every global isometry is bijective.
2. Show that every isometry is an embedding and therefore every global isometry is a homeomorphism.
3. Find a function that is a homeomorphism but not an isometry.